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Using Newton's Laws to Determine How Forces Affect an Object's Motion

Now that we have discussed how to find the net force, Newton's laws will tell us how that net force affects an object's motion. Sir Isaac Newton is credited with, among other things, inventing calculus as well as discovering three laws that govern all motion on Earth (except for on the molecular level) in the late 17th century. These three laws tell us how forces affect an object's motion. We'll address them one at a time.

Newton's First Law specifies what happens when the net or total force on an object is 0. If all the individual forces cancel each other out, there should be no change in the motion. A perfect example of this law is you sitting in your chair right now. Your weight is pulling you down, but the normal force of the chair cancels that out, so there's nothing left to force you to move.

Newton's First Law

If Fnet = 0, there's no change in motion.


In other words, if an object is at rest, it stays at rest until a net external force changes it. Likewise, if an object is moving, it continues at the same speed in the same direction until a net force changes it. At this point, you might be thinking, "But if I roll a ball across the floor, it slows down and eventually comes to rest." That's true. But the net force is not 0. Don't forget about friction; that is the net external force responsible for slowing down the ball.

Example 11.7: Newton's First Law on Air Hockey

Suppose you're coding an air hockey game. An ideal table should have no friction. A player hits the puck and gives it an initial velocity of [10 5]. As the puck approaches the wall 3 seconds later, what is its current velocity just before it hits the wall?

Solution

If the game takes place on a good air hockey table with no friction, there's no net external force. Therefore, the puck will still be moving with a constant velocity of [10 5].

NOTE

In Example 11.7, the puck changes its motion only when it hits the wall or the other player's paddle.


Newton's First Law tells us what happens when the net force is 0. At this point, are you wondering what happens when the net force is something other than 0? You already know that a net external force changes an object's motion, but by how much? That's precisely what Newton's Second Law tells us.

Newton's Second Law

Fnet = ma

where m = mass and a = acceleration.


NOTE

Note that when the net force is 0, the acceleration must be 0, which is exactly what Newton's First Law says.


This law tells us several things. First, it shows that the more force you place on an object, the faster it speeds up. It also tells us that if two objects receive the same amount of force, the smaller object speeds up faster than the larger object. Most importantly, Newton's Second Law lets you numerically calculate an object's acceleration based on its mass and the net force.

Example 11.8: Newton's Second Law

In Example 11.6, you found that the net force on the sled was [44.6 0]. What's the resulting acceleration?

Solution
  1. Remember that the loaded sled weighs –355.9N. Use that to calculate the mass:

    w = mg

    –355.9N = m(–9.8m/s2)

    m = 36.3kg

  2. Use the net force and mass to calculate the acceleration:

    Fnet = ma

    graphics/11equ07.gif


After you calculate the acceleration, you're right back to the five equations of motion from Chapter 10, "Motion in Two and Three Dimensions." The most important thing to remember about Newton's Second Law is that the net force can change at any time. An object accelerates only when the net force is applied. When the force stops, the acceleration stops, and the velocity becomes constant (Newton's First Law). Fortunately, in game programming everything is recalculated on a frame-by-frame basis, so as soon as the net force stops, the next frame is based on 0 acceleration. So overall, the process starts with the forces, which determine acceleration, which determines velocity and displacement. Ultimately, you need to track where an object should be located as time is incremented.

Example 11.9: Newton's Second Law with the Five Equations of Motion

Suppose you're coding a game with a top-down view, and the player needs to roll a large boulder into a stand in order to unlock a door. The boulder is sitting still, and it weighs 50lbs. If the player pushes with a constant force of 400N @ 30°, and the coefficient of kinetic friction is 0.25, how far will the boulder move after 1, 2, and 3 seconds?

Solution
  1. Start with a free-body diagram to organize all the forces. This is done in Figure 11.7.

    Figure 11.7. A free-body diagram for a boulder.

    graphics/11fig07.gif

  2. Numerically calculate FK:

    w = –50lbs = –222.42N

    N = 222.42N

    FK = –mKN = –0.25(222.42N) = –55.605N (opposite of FP)

  3. Calculate the net force. Remember that all forces must be added in component form, so break both forces into components:

    FP = 400N @ 30° = [346.41 200]

    FK = 55.605N @ 210° = [–48.16 –27.80]

    Fnet = FP + FK = [298.25 172.20]

  4. To use Newton's Second Law, you need to know the boulder's mass:

    w = mg

    –222.42N = m(–9.8m/s2)

    m = 22.7kg

  5. Use the net force and mass to calculate the acceleration:

    graphics/11equ08.gif


  6. Now that you know the acceleration, you can use the five equations of motion just like before. If the boulder starts from rest, vi = [0 0]. If you know the acceleration, initial velocity, and time, you can use the 4th equation of motion to find displacement:

    graphics/11equ09.gif


  7. Use that equation to solve for displacement for each of the three times:

    graphics/11equ10.gif


Even though Newton's Third Law does not appear often in code, this discussion would be incomplete without it, so let's quickly mention it. You might have heard the saying "For every action there's an equal and opposite reaction." Newton's Third Law basically says that for every force there's an equal and opposite force. For example, if I punch my opponent in a fighting game, I am putting a force on that person. At the same time, that person's body is exerting a force back on my fist, which is why my fist slows down (and why it hurts!)

Newton's Third Law

For every force there is an equal and opposite force, or, when two objects come into contact, they exert equal and opposite forces upon each other.


This chapter actually took a step backwards in the overall framework of a physics simulation. Before you can track the motion of an object, you must examine what causes the motion, and that means forces. After you have brainstormed all the forces acting on the object being modeled, you can add them up to find the net force. If the net force is 0, the object continues to move with a constant velocity. If the net force is nonzero, you can use the mass to numerically calculate the acceleration, which then determines how fast and how far the object moves.

Self-Assessment

1.

If you lay this book on a table, and nothing touches it, where should it be 2 minutes later?

2.

If you're in the space station, where there's no friction, and you toss a pen to your partner with an initial velocity of 5m/s @ 45°, what's its velocity when it reaches your partner?

3.

What is the acceleration for the object in the free-body diagram shown in Figure 11.8?

Figure 11.8. The free-body diagram for question 3.

graphics/11fig08.gif

4.

If the forces listed in question 3 are applied for 5 seconds, and the object starts from rest, how far will the object move?

5.

Suppose you're coding a game with a top-down view, and the player needs to push a cart full of treasure back to home base. The cart is sitting still, and it weighs 35lbs. If the player pushes with a constant force of 200N @ –60°, and the coefficient of kinetic friction is 0.05, what's the cart's acceleration?

6.

If the player in question 5 pushes the cart with a constant force for 3 seconds, what's the cart's displacement?


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